(dx)/(dt)=k^1(1-x)+k^2(1-x)^2

解

\frac{d x}{d t} = k^{1} \cdot (1 - x) + k^{2} \cdot (1 - x)^{2}

x = \frac{e ^{k t + c_{1}}}{- 1 + e ^{k t + c_{1}}} - \frac{1}{- 1 + e ^{k t + c_{1}}} - \frac{1}{k ( - 1 + e ^{k t + c_{1}} )}

解答ステップ

\frac{d x}{d t} = k^{1} \cdot (1 - x) + k^{2} \cdot (1 - x)^{2}

分離可能 ODE を解く:

x = \frac{e ^{k t + c_{1}}}{- 1 + e ^{k t + c_{1}}} - \frac{1}{- 1 + e ^{k t + c_{1}}} - \frac{1}{k ( - 1 + e ^{k t + c_{1}} )}

x = \frac{e ^{k t + c_{1}}}{- 1 + e ^{k t + c_{1}}} - \frac{1}{- 1 + e ^{k t + c_{1}}} - \frac{1}{k ( - 1 + e ^{k t + c_{1}} )}